I would venture to say this is a difference between REAL userid and EFFECTIVE userid for a given process and how exactly sudo operates on each of your two machines.
MINT appears, when sudoing a process, to change both the effective user id (root) and the real user id (the user who invoked the process) to root. Thus the original user cannot kill either the sudo process or the sudoed process (gedit). Xubuntu (and, in my case, Fedora) do not do this. The real userid is still the original user.
Use the ps command to see process owned by users (real or effective). I ran your sudo gedit command.
ps uaxf produced this tree:
myuserid 22868 0.0 0.0 163400 4872 pts/2 Ss 13:19 0:00 | \_ -zsh
root 30392 0.0 0.0 205044 3736 pts/2 S+ 13:45 0:00 | | \_ sudo gedit
root 30417 0.0 0.1 699788 23692 pts/2 Sl+ 13:45 0:00 | | \_ gedit
Clearly the effective user id for both sudo and gedit are root, and not myself. Then I used ps with the --User option (--User userlist Select by real user ID (RUID) or name.) to see who the real userid for the processes are:
ps --User "myuserid"
This produced a lot of output, but there was one line that stood out:
30392 pts/2 00:00:00 sudo
But the process 30417 was conspicuously missing...
Then I ran:
ps --User "root"
This also produced a lot of output, but this line stood out:
30417 pts/2 00:00:00 gedit
But process 30392 was absent.
So, It seems that eventhough the sudo process (30392) has an effective owner of root, the real owner is me, and I'm able to kill it. The gedit process, on the other hand, has both effective and real ownership by root and is not (directly) killable by me. I'd guess that on Lint, both process are both effectively and really owned by root.
I thought this might be affected by the stay_setuid option in /etc/sudoers, but the description of that doesn't really make this seem likely.
Are there any differences in your /etc/sudoers files between Mint and Xubuntu?